We’re still motoring through numeric waters but hang in there; the shore is just ahead. This is the last math theory post… for now. I do have one more up my sleeve, but that one is more of an overly long (and very technical) comment in reply to a post I read years ago. If I do write that one, it’ll be mainly to record the effort of trying to figure out the right answer.
This post picks up where I left off last time and talks more about the difference between numeric values and how we represent those values. Some of the groundwork for this discussion I’ve already written about in the L26 post and its follow-up L27 Details post. I’ll skip fairly lightly over that ground here.
Essentially, this post is about how we “spell” numbers.
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3 Comments | tags: base 10, base 2, base 8, Frederik Pohl, Heechee, irrational numbers, Leopold Kronecker, natural numbers, number bases, number names, numbers, pi, prime numbers, rational numbers | posted in Math, Sideband
In this post I’ll show how Set Theory allows us to define the natural numbers using sets. It’s admittedly a very abstract topic, but it’s about something very common in our experience: counting things. Seeing how numbers are defined also demonstrates (contrary to some false notions) that there is a huge difference between a number and how that number is “spelled” or represented.
Note: I am not a mathematician! This topic is right on the edge of my mathematical frontier. I wanted this addendum to the previous post but be aware I may misstep. I welcome any feedback from Real Mathematicians!
But go on anyway… keep reading… I dare ya!
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8 Comments | tags: counting, counting numbers, natural numbers, numbers, set theory, successor function | posted in Math, Sideband
Be warned: these next Sideband posts are about Mathematics! Worse, they’re about the Theory of Mathematics!! But consider sticking around, at least for this one. It fulfills a promise I made in the Infinity is Funny post about how Georg Cantor proved there are (at least) two kinds of infinity: countable and uncountable. It also connects with the Smooth or Bumpy post, which considered differences between the discrete and the continuous.
This first one is pretty easy. The actual math involved is trivial, and I think it’s fascinating how the Yin/Yang of separate units versus a smooth continuum seems a fundamental aspect of reality. We can look around to see many places characterized by “bumpy” or “smooth” (including Star Trek). (The division lies at the heart of the conflict between Einstein’s Relativity and quantum physics.)
So, let’s consider Cantor.
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10 Comments | tags: Cantor, Cantor's Diagonal, finite, Georg Cantor, infinity, integers, irrational numbers, natural numbers, numbers, rational numbers, real numbers | posted in Math, Sideband
I didn’t realize it at the time, but by staying up reading until 5 AM this morning, I was awake for the ironically named “Beginning of Summer.” I say “ironically” because the Summer Solstice is the point when the days start to get shorter again. The beginning of summer is also the beginning of the darkness.
Which means that my pagan side mourns the Summer Solstice as much as it celebrates the Winter one. These days, it’s hard not to see a larger parallel in society. Many of us feel and fear society is sliding into darkness — inexorably spinning along a path towards a Winter of Disaster.
This Solstice, as food for thought, I want to introduce The Five E’s…
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10 Comments | tags: darkness, economics, education, entertainment, environment, light, media, punching, rock bottom, social issues, social media, Solstice, summer, Summer Solstice, Sun, Supreme Court, W.G. Sebald | posted in Rant
Voices. It begins with voices. Even before we are born, we hear voices. Human language is the most complex form of inter-species communication that we know. It takes years to learn and many more years to become fluent. Mastering it takes serious dedication and practice.
In the public square, it also begins with voices. The voices of men filled it first. As time marches on other voices are raised: the voices of women; the voices of nationality and race; political voices; religious voices; gay voices; vegan voices and more. Now the public square is filled with the dynamic clamor of many different voices.
To go beyond the beginning, we must listen to the voices.
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12 Comments | tags: listen, logos | posted in Writing
Earl Grey. Hot!
I’ve written about the Yin-Yang of analog versus digital, a fundamental metaphor for how reality can be smooth or bumpy. I’ve applied the idea to numbers, where we see two types of infinity — countable (discrete, digital, bumpy) and uncountable (continuous, analog, smooth). There is also how chaos mathematics says that — the moment we round off those smooth numbers into bumpy ones — our ability to use them to calculate certain things is forever lost.
I’ve also written about Star Trek replicators and transporters, as well as the monkey wrench of the hated holodeck. According to canon, all three use the same technology (which raises some contradictions for the holodeck).
Today, for Science Fiction Saturday, I want to tie it all together in another look at transporters and replicators!
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12 Comments | tags: analog, Captain Kirk, Captain Picard, chaos theory, digital, Earl Grey, infinity, replicators, Star Trek, transporters, Yin and Yang | posted in Sci-Fi Saturday
About 500 years ago a thing happened in Europe: The Scientific Renaissance. It was part of a larger thing, called the Scientific Revolution. These were the seeds that lead to the Age of Enlightenment, when science and rationality were the saviors of humanity lifting us up from the dark ages.
Now the Renaissance is mostly seen as a traveling annual party where people can play Medieval dress-up and eat giant turkey legs (thus proving that anything can be trivialized and you are what you eat). Which is all fine. I enjoy a good outdoor party as much as anyone, and it is interesting finding out what mead actually tastes like.
But I fear we’re forgetting the advances made in the real Renaissance and setting sail back to the Dark Ages.
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Leave a comment | tags: Age of Enlightenment, rational thought, Renaissance, Scientific Renaissance, Scientific Revolution, The Atomic Age, turkey legs | posted in Rant
In 2006 Mike Judge (Office Space, King of the Hill) wrote and directed a film called Idiocracy. It postulates a future 500 years hence when, due to “The Marching Morons” theory, the world is entirely populated by extremely stupid people. In this dystopian future, advertising and commercialism have run rampant in an anti-intellectual culture devoid of intellectual curiosity and thought.
As I watch what passes for communication or discussion on the interweb, as I watch in horrified fascination at the complete failure of nuanced — let alone deep — thought in people today, I begin to realize one thing:
The Idiocracy is here; I live in a world filled with morons.
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24 Comments | tags: Cyril M. Kornbluth, education, Idiocracy, idiots, ignorance, intelligence, interweb, Mike Judge, morons, stupid, stupid people, stupidity, The Marching Morons | posted in Rant
I’ve written several times about the many places we see the idea of a Yin and Yang duality played out in the real world. Even the application of the Yin and Yang concept has a Yin (of true opposites) and a Yang (of thing and not-thing). For example, the opposite of light is not-light, but the opposite of positive is negative.
One of the true opposites is the idea of analog versus digital or, more generally, of continuous versus uneven. Recently I was thinking about the differences between various sports, and I realized there’s a connection to the “smooth or bumpy” distinction I wrote about a while back. Looked at in terms of play, some sports are essentially continuous while others are not.
It turns out that some sports are “analog” while others are “digital.”
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7 Comments | tags: Alfred Hitchcock, analog, basketball, continuous, digital, discrete, football, hockey, MacGuffin, Pulp Fiction, Quentin Tarantino, sports, team sports, tennis, Yin and Yang | posted in Baseball
Happy Twins; Happy Fans!
The Minnesota Twins have had three of the worst years (2011-2013) in franchise history, but this year there have been hints that this year might be different. In April they had a win-loss record of 12-11 (.522) and averaged 5.57 runs per game. The downside was that starting pitchers had a 6.02 ERA.
In May, the pitching got better; starters threw a very good 4.03 ERA. Unfortunately, the bats went ice-cold. Runs per game were down to 3.17, and Twins lagged behind their opponents by 27 runs. The month ended with a 13-16 (.448) record.
June is off to a bang with one of the most exciting wins the Twins have seen in years!
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5 Comments | tags: Brian Dozier, Eduardo Nunez, Josh Willingham, Minnesota Twins, MLB, New York Yankees, Oswaldo Arcia, Phil Hughes | posted in Baseball
Logos con carne 